Recently purchased said bike on CL for 120$ owner said it was purchased in mid 2011

The rear section was pretty gunked up. After cleaning and alignment seems almost right.
However I do notice some chain skips that arent going away.

Id prefer to just get a new chain , and Cassette/Freewheel then go nuts trying to pinpoint an issue. Problem is since it was bought off CL Im not entirely sure the exact parts I would need.

Can someone please give me advice on parts here ?

Current model info on BD says
Cassette 13-28T or 12-28T (no choice) 7 Speed Hyperglide (21 gears total)

Would any of these be a suitable replacement ?
Cassette
Amazon.com: Shimano HG41 7 Speed Mountain Bike Cassette: Sports & Outdoors

Freewheel

Amazon.com: Shimano Tourney 7Spd 14-28T Freewheel: Sports & Outdoors

Amazon.com: Shimano MF-HG37 Tourney Freewheel (13-28T 7 Speed): Sports & Outdoors